PROPERTIES OF A UNIVERSAL WAVE FIELD
(pv^{1} = const.; p/v = const. = RT)
Summary:
A
theoretical gas ( the Chaplygin/Tangent adiabatic gas) is currently being studied as an exotic
cosmic fluid in cosmology to help
explain dark energy, dark matter and an apparent speedup in the expansion of
the universe.
Here we
show that this exotic cosmic fluid has isothermal
as well as adiabatic motions, and so is actually a universal compressible
field (UF) with unique properties.
These
unusual properties include (1) the unique ability among known fluids or gases to propagate stable waves of
any strength, which also (2) uniquely
obey the simple classical wave
equation, (3) an ability to support transverse waves, which is something
impossible for any other known or theoretical fluid, and which thus provides
for the first time a physical basis for the existence of the electromagnetic
field and of transverse electromagnetic
waves, and for the transfer of radiation through space at the
speed of light, (4) the ability to carry gravitational waves which transfer
gravitational force through space, and finally, ( 5) it may also serves as the basis for the transfer of quantum information
through space at quasiinstantaneous
speeds.
Contents
1.
Introduction
2.
Dynamics and Thermodynamics of Compressible Exotic Fluids ( Chaplygin gas and
Tangent gas) with k = −1
2a.
Archive Insert: Thermodynamic Properties of an Isothermal Gas Law for the
Chaplygin/tangent Field
3.
A Universal Field (UF)
4.
A Universal Field (UF) which Supports Transverse Waves and the Transmission of
Transverse Electromagnetic Waves through Space
5.
A UF Which Supports the Transmission of Gravitational Force through Space
6.
A UF Which Supports the Transmission of Quantum Information though Space
7.
Experimental Evidence for the UF
1. Introduction
Is There a Universal, Cosmic, Compressible Wave Field ?
The concept of a universal
physical entity filling all space in the cosmos and serving to transmit
electromagnetic and gravitational forces between material bodies has had a long
and interesting history [1]. Although the concept existed in the ancient
classical era in a prescientific form, it was Descartes who first formulated
it scientifically. His ‘ether’ was
incompressible, so that its light waves would need to travel instantaneously.
Consequently, it was not long before Fermat pointed out that experiments
showed, instead, that light traveled through space at a finite speed.
However, evidence
that wave motions were undoubtedly associated with light advanced under Huygens
and Young restored the concept of the wavecarrying ether, and the task of reconciling the physical properties of this
hypothetical ether with experimental
reality was again taken up. The two greatest difficulties were the known finite
speed of light ( 3 x 10^{8} m/s),
and the impossibility of any conceivable fluid being able to support the
transverse vibrations corresponding to light waves.
Maxwell’s
successful, transverse electromagnetic wave theory of light ended any lingering
controversy as to the transverse nature of light waves, but did little to support
the ether in its problems. In fact, the elastic ether quickly became replaced
by the concept of a field with a dielectric constant and a magnetic
permeability. Almost coincidentally,
difficulties in attempts to use classical methods of adding velocities of
material bodies to the speed of light took front and centre as the negative
results of the optical experiments of
Michelson, Morley Miller, Joos and
others were evaluated. These tests failed to show the correct orbital speed of
the earth around the sun using the classical theory of the flow of light waves
through an ether. The experimental
variations observed when the optical interferometers were rotated
through 90 degrees were only about onetenth
as large as predicted by the classical ether theory, and
this result eventually was interpreted as meaning that it was impossible
to use light to detect any motion at all
through space. Other optical experiments such as the Fizeau Effect and the
Sagnac Effect which did show large experimental effects were eventually
ascribed to different causes.
This led to the
Poincare/Fitzgerald /Lorentz / Einstein work on relative motion, and
eventually the whole idea of an ether or
wave medium which supported light waves
was abandoned. Today, when compressible
flow theory has been fully developed, this abandonment appears to have perhaps
been too hasty, and a recent review of the experimental evidence from the
standpoint of compressible flow as
opposed to the old classical incompressible flow ( with its direct addition of
velocities ( c +V) and (cV) ) is
available at website www.energycompressibility.info [ Appendix A: Compressible
Flow and Results of MichelsonMorley Type Experiments].
In any case, the
rise of quantum theory with its photon “particle” would probably have
eventually abolished the old classical ether concept even without Michelson and Morley and their
successors.
Historically, the
ether was dead. although the problems with light were not thereby resolved. Gas
dynamics, high speed atmospheric science, aeronautics and other aspects of
compressible flow theory were just emerging, and so the important and
significant fact that compressibility might explain the observed finite
speed of light never came to the fore although it was tentatively advanced as a
possible help in the solution to the MichelsonMorley dilemma by Lorentz
himself.. Instead, the quite radical,
essentially algebraic and geometrical theory of relativity and its Lorentz
invariance took over, and it has served well enough as a fairly close
approximation to the experimental results.
The application
of compressible flow theory to relativity, electromagnetism and cosmology began
in the 1980’s [2, 3, 4, 5] and the new approach immediately showed numerous
successful applications to problems in these fields. Now the concept is being
widely applied to the problem of the observed acceleration of expansion of the
cosmos by the introduction of an exotic cosmic compressible theoretical fluid
based on the so called Chaplygin gas [6  12].
We now proceed to
examine more deeply the thermodynamic and wave properties of exotic compressible fluids and their possible
interactions with our physical world.
2. Dynamics and Thermodynamics of Exotic Compressible Fluids ( Chaplygin and
Tangent gases) with k = 1)
Subjects:
Basic Equations of Compressible Fluid Flow
Adiabatic Equations of State for the
Chaplygin gas and Tangent gas (k = 1)
A New Isothermal Equation of State for this
Exotic Fluid ( k = 1)
2.1 To describe the motions of any compressible fluid, three basic equations are needed:
1.Euler’s classical hydrodynamic equation
of motion;
For 1dimensional
flow this equation is
∂u/∂t + u ∂u/∂x +
v ∂u/∂y +w ∂u/∂z = −(1/ρ) ∂p/∂x
(2.1)
where the term on the right hand side is called the pressure gradient force.
2. The equation of
continuity, or conservation of mass
∂ρ/∂t + div (ρw) = 0
(2.2)
3. Equation of State relating pressure p
to density ρ ( or to its reciprocal,
the specific volume v = 1/ρ)
(a) For real
physical gases undergoing adiabatic
motions ( i.e.(no heat flow, dQ = 0) the general equation of state
is :
pv^{k} = constant (2.3)
p/ρ^{k} = constant
(2.3a)
and the wave
speed is c^{2} = kpv, where k, the ratio of the specific heats (k = c_{p}/c_{v}
) is the adiabatic exponent (sometimes
also denoted as γ). The adiabatic exponent ratio k is also related to the
number of ways n that the energy of the
system is divided as k = (n+2)/n; n = 2/(k−1).
(b) For isothermal motions in a
real gas, the equation of state becomes
pv = RT or p/ρ
= RT
(2.4)
and the wave speed now is given by c^{2} = pv.
For real gases
the equations of state all lie in
Quadrant I of the pressure –volume field of Figure 1.
Figure 1. Pressurevolume in Compressible Fluids
Quadrant 1: Real gases and Tangent gas (exotic)
Quadrant IV. :Chaplygin
gas and Tangent gas (exotic gases)
One additional
relation is very useful, namely the kinematic energy flow equation relating
compressive wave speed c to relative flow velocity V:
c^{2} = c_{o}^{2} – V^{2}/n
(2.5)
where n ( n
= 2/(k – 1)) is the number of ways the energy of the system is
partitioned. If we divide through by the
square of the static wave speed c_{o}^{2} we get the wave speed
ratio
c/c_{o} = [ 1 – V^{2} / n c_{o}^{2}
]^{1/2} (2.6)
which ( when n = 1 ) is formally identical to the
Lorentz/Fitzgerald contraction factor of special relativity theory (Section 4).
For unsteady or
pulsed flow, the energy equation (2.5) becomes
c^{2} = c_{o}^{2} – V^{2}/n – 2cV/n where the additional, or ‘pulse’ term 2cV/n is of great interest to quantum
phenomena as we shall see in Section 6.
2.2 Adiabatic Equations of State for Exotic
Fluids where k= −1
For real gases
and fluids the adiabatic exponent k in Eqs. 2.3
and 2.3a is always positive. However, if
k is, instead, taken as being a negative number then the properties of the resulting theoretical
fluid change radically. In 1901 a Russian aerodynamicist, S.A. Chaplygin [10]
first proposed such a purely theoretical,
compressible fluid  now called the
Chaplygin gas and having k = −1  to help calculate certain features of jet flow in gases. Since
his theoretical gas does not actually exist in our known physical world, it has
had little application, except for simplifying some calculations in
aerodynamics.
Within the last
five years, however, the cosmological problem raised by an unexplained
acceleration in the expansion of the universe
has been cited by some cosmologists [6,7, 8, 9] as indicating that the
Chaplygin gas may exists as an exotic universal “cosmic fluid” which is the
physical seat of the socalled ‘dark
energy’ of the universe, presently calculated to comprise about 74% of the
total ‘matter’ of the cosmos.
The Chaplygin gas [10  15] has the
adiabatic equation of state
pv^{1}= p/v = p ρ = constant, or
p = −Av = −A/ρ
(2.7)
where p is the
pressure, v = 1 /ρ is the specific
volume, ρ is the density per unit
mass of fluid and A is a positive
constant. This equation plots with negative slope, dp/dv, on the
pressurevolume diagram ( Fig. 1) and
its pressure p is always negative. This negative pressure is the attractive
feature for the present day cosmologists who are concerned with the apparent
accelerated expansion of the universe, and the Chaplygin gas is increasingly
proposed as a physically real exotic cosmic fluid to address this cosmological
problem. However, its properties are quite
bizarre compared to our real world gases and the success of this innovation in
rescuing general gravitation and superstring theories of gravity is
problematical. The Chaplygin gas lies entirely in Quadrant IV of Figure 1.
A closely related
exotic fluid called the Tangent Gas has an equation of state
identical to the Chaplygin gas except for the addition of a constant B
which makes its pressure positive for values of v less than
B. It may lie in either Quadrant 1 or
Quadrant IV of the pressurespecific volume diagram of Figure 4. Its equation of state is
pv^{1} = constant or
p = −Av + B = −A/ ρ +B
(2.8)
In this form, the
relationship became very useful to aerodynamics in the 1940’s [14]. As a
tangent curve to the adiabatics and isotherms for atmospheric air, the “tangent
gas” provided a linear relationship between pressure and density, which, for
small variations of these variables, gives a simple approximation to the more
cumbersome, exact, nonlinear thermodynamic equations for any compressible
fluid except the UF.
2.3 A New
Isothermal Equation of State for a Compressible Fkuid with k = −1
A
generalized adiabatic equation of state
for the fluid with k = −1 is pv^{k } =constant, and so this becomes
p v^{1} = p/v = const. = ± A, or
p = ± Av
(2.9)
We now must
choose the sign before the positive
numerical constant A, and this will
determine the slope of the equation of state in the pv field. In the case of
the Chaplygin gas ( Eq. 2.7) the sign is
chosen as negative so as to give p = −Av = −A/ρ, and for the Tangent gas (Eq. 2.8) as p = −Av + B = −A/ ρ +B , mainly
because this makes them both agree with all real gases in having egative slope − (dp/dv) on the pv diagram, and also guarantees a positive wave speed with c^{2} = +(dp/dρ).
However, there is
also no apparent reason to completely reject the alternative choice of +A for
the equation of state, that is, to chose a positive slope for dp/dv. This would
give the isothermal equation of state for the fluid:
p =+ Av
(2.10)
and
p = +A/ρ –B
(2.10a)
These two new isothermal curves with positive constant
A are strictly orthogonal to the
adiabatic Chaplygin and Tangent gas curves. They constitute the desired isothermal
equation of state for the fluid with
k = −1, since
p/v =+A, and, if the positive
constant A is set equal to a constant temperature + A =RT ,we would then
have p/v = RT which is certainly an isothermal relationship.
We then have :
Real gases ( k ≥
1) : pv>^{1} = constant = +A
is the general adiabatic equation of state
Real gas ( k = 1)
: pv^{+1} = constant. = +A =
RT is the ‘isothermal gas’, and is also the equation of state for any
ideal gas with constant
temperature T.
Exotic gases ( k
= − 1) :
a)
Chaplygin gas: Adiabatic equation of state p = −Av = −A/ρ (Eq. 2.7)
Isothermal equation of state p
= Av = vRT; p/v = RT (Eq. 2.10)
b) Tangent Gas
Adiabatic
p = −Av + B = −A/ρ
+ B (Eq. 2.8)
Isothermal
p = Av − B = vRT −
B (Eq.
2.10a)
We now propose to treat these exotic states, not as
separate physical entities or ‘gases’ but instead as simply the
adiabatic and isothermal equations of state of one single, universal
compressible field ( UF) whose adiabatic exponent is k = −1.
Figure 2. Equations of State for the Universal Field (k = − 1; pv^{1} = const.)
2.a (Archived Insert) May/06
Thermodynamic properties of an isothermal gas law for
the Chaplygin/tangent field
Bernard A. Power^{a)}
Montreal, Quebec, Canada
March, 2006
An isothermal equation of state is formulated to match the adiabatic Chaplyin/tangent gas, and it appears to be a general gas law for a whole field whose thermodynamic properties are very unusual. In this field a working substance expands with cooling and contracts with heating, and both processes take place without any work being done. The 2^{nd} Law is inapplicable.
The theoretical or exotic fluid known as the Chaplygin/tangent gas has an isentropic/adiabatic equation of state which has been widely studied and applied in aerodynamics and fluid dynamics for many years.^{1,2,3,4,5,6} However, the adoption by cosmology of the negative pressure attribute of the Chaplygin gas as a possible solution to the observed increased expansion of the cosmos is only recent. ^{7,8,9} A potential role as a universal cosmic fluid, however, would seem to merit an examination of all aspects of this field.
The equation of state for the Chaplygin gas is the linear relationship p = − Av, (pv^{1} = −A) ; it lies in Quadrant IV of the pv diagram where it always has negative pressure. The dynamically identical Tangent gas has an added constant and is also linear, with p = − Av + B, ( pv^{1} = −A + B] ; in Quadrant I it has positive pressure (Fig. 1).
The Chaplygin/tangent gas can also be described by the adiabatic equation of state pv^{k } = −A, where A is a positive constant, k has the value of − 1, and where the minus sign preceding the constant A provides a desired negative slope dp/dv on the pressurevolume diagram.
The proposed isothermal equation of
state for this field is p = +Av corresponding to the adiabatic Chaplygin gas, and p = +Av –B
corresponding to the adiabatic tangent
gas. If the
positive constant A is multiplied by a
constant temperature T_{c}, we would then have p/v = A (T)_{c}
, p = vAT_{c}, and p = vAT_{c} − B, which provide the desired isothermal
relationship, provided that A takes on
the proper dimensions. The isotherms in
pressure and volume are seen to be
strictly orthogonal to the adiabatics of the Chaplygin and Tangent gas. (Fig. 1)
Since v =
1/ρ , these linear isothermal relationships in pressure and volume ( p =
vA T_{c} , etc ) can also be
written hyperbolically in pressure and specific density ρ as
pρ =
AT (1)
The ideal gas law ( pv = RT) is hyperbolic in pressure and volume, but linear in pressure and density ( p = ρRT). The proposed new isothermal equation of state for the k = −1 field is the reverse, being linear in pressure and volume and hyperbolic in pressure and specific density.
It would appear that the new relationship is not so much just an isothermal counterpart to the Chaplygin/tangent gas, but rather that it plays the more fundamental role of being the general gas law of the field, with the Chaplygin and tangent cases being simply the exceptional modifications to the gas law that apply under the adiabatic condition ( dQ = 0), just as is the case with real adiabatic gases which depart exceptionally from the ideal gas law. The term universal field ( UF) would embrace both the isothermal and adiabatic equations and where “isothermal” equation we have derived would become the basic gas law ( pρ = AT )for the whole field.
The thermodynamic properties of this UF are very unusual because of (1) the direct proportionality of pressure and volume, (2 a negative specific heat either at constant volume or at constant pressure and (3) the positive slope +dp/dv of the isotherms on the pressurevolume diagram and its effect on expansion or contraction..
First, the isothermal pressure increases with volume expansion is the inverse of that for the ideal gas , so that heating now accompanies contraction and cooling brings about expansion.
Second the entropy relationships in the UF are also unusual since we have k = c_{p}/c_{v} = −1, so that either c_{p} or c_{v} must be negative. In the ideal gas with T and V as independent variables the entropy change is ∆S = c_{v} ln (T_{2}/T_{1}) + R ln(V_{2}/V_{1}). And with T and P as independent variables, it becomes ∆S = c_{p} ln (T_{2}/T_{1}) – R ln(P_{2}/P_{1})
In the UF we have dS= c_{v} dT + (∂P∂T)_{v} and, from P =AvT we have (∂P∂T)_{v} = Av^{2}/2 so that we obtain the corresponding entropy changes in the UF as ∆S = c_{v} ln (T_{2}/T_{1}) + Av^{2}/2 and ∆S = c_{p} ln(T_{1}/ T_{2}) ) −Av^{2}/2
(In the UF the magnitude of the reversible heat Q_{r} needed to calculate the entropy change ∆S is readily visualized on the pressure volume diagram, since the Carnot cycle is then depicted simply as the rectangle formed by the appropriate linear adiabatics and isothermals, and Q_{r} is the area enclosed).
Third, the positive slope +dp/dv of the isothermals on the pressurevolume diagram implies that there is no resistance to pressure/volume increases or decreases, which would therefore take place at ever increasing speed following any initial impulse.. This is analogous to an expansion into a vacuum in the usual analysis of an isothermal process in an ideal gas where the work pdV is then zero In the UF in an isothermal process it would seem that this work would also necessarily approach zero. This would mean complete reversibility of all UF processes, both adiabatic ( Chaplygin/tangent) and isothermal, with no referred direction for any physical change. All complete cycles in the UF would appear to be isentropic. Heat would be convertible into internal energy but not into work.. The 2nd law of thermodynamics would not apply. Clearly the properties of this unusual field should be further explored and critically evaluated.
References
1.. S. A.
Chaplygin, “On Gas Jets,” Sci. Mem.
2. H.S. Tsien, “TwoDimensional Subsonic Flow of Compressible Fluids,” J. Aeron. Sci. 6, 399 (1939).
3. T.Von Karman, “Compressibility Effects in Aerodynamics,” J. Aeron. Sci. 8, 337 (1941).
4. A. H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow. 2 Vols.( John Wiley & Sons, New York, 1953).
5. R. Courant and
K. O. Friedrichs, Supersonic Flow and Shock Waves. (Interscience , New York, 1948).
6. Horace Lamb, Hydrodynamics. 6^{th} ed . ( Dover Reprint, Dover Publications Inc.
7.
N.A. Bachall, J.P. Ostriker,
8 A. Kamenshchick, U. Moschella and V. Pasquier, “An alternative to quintessence,” Phys
Lett. B 511, 265 (2001).
9. N. Bilic, G.B. Tupper and
R.D.Viollier, “Unification of Dark Matter
and Dark Energy: The Inhomogeneous Chaplygin Gas,” Astrophysics,
astroph/0111325 ( 2002).
Fig. 1. Equations of State in the UF
3. A Unique, Universal
Field (UF) 1 Which
Can Support Stable Longitudinal and Transverse Waves of Any Amplitude
The
evidence seems to be that instead of
there being three separate exotic
“gases”( Chaplygin gas, Tangent gas and Orthogonal or isothermal gas) that
there is only a single, compressible, fluid entity or field having the usual separate adiabatic and
isothermal equations of state.
Equations of State
Kinematic Energy Equation for Relative
Motion
Force in the UF
Wave Motions in the UF
Isentropic ratios of p, ρ and T in the
UF
Static pressure, density and temperature
values in the UF
We now explore
this proposition that there exists a
single Universal Field (UF)which has the
compressible properties of the adiabatic
Chaplygin/Tangent gas and the new isothermal
orthogonal gas, and which supports stable waves obeying the classical wave
equation. In succeeding Sections this will then be expanded to show that this
Universal Field is also the physical entity which supports and transmits (a)
light and other transverse electromagnetic waves, (b) gravitational force
between masses, and (c) transfers
quantum information through space.
Some of its
principal physical properties are summarized as follows:
3.1 Equations of State relating pressure , p, to specific
density ρ ( = 1/v) and where n =−1
and k = (n+1)/n = −1:
Adiabatic: p = −Av
+ B = −A/ρ +B (Chaplygin/Tangent gases)
(3.1)
Isothermal:
p= +Av −B ;
pv = RT (Orthogonal Gas)
(3.2)
3.2 Kinematic Energy Equation : ( relating wave energy c^{2} to
kinetic or flow energy V^{2}, and wave speed c to relative motion V)
c^{2} = c_{o}^{2} – (1/n) V^{2
} (3.3)
and, since n = −1
we have
c^{2} = c_{o}^{2}
+ V^{2
}(3.4)^{ }
If we divide
through by the static wave speed c_{o}^{2} we get
c/c_{o} = [ 1 – V^{2} / n c_{o}^{2}
]^{1/2
}(3.5)
which ( when n = +1 ) is the
/Fitzgerald/Lorentz contraction factor of special relativity theory to
be mentioned in Section 4.
3.3
Force in the Universal Field (UF)
In the UF as in any compressible fluid the
force is given by the Euler equation ( Eq. 1).in Section 2.
For 1dimensional
flow this equation is
∂u/∂t + u ∂u/∂x +
v ∂u/∂y +w ∂u/∂z = −(1/ρ) ∂p/∂x
(2.1)
where the term on the right hand side is called the pressure gradient force.
3.4 Wave Motion in the UF
The Universal
Field in unique in that (1) it is the
only field in which the Classical Wave Equation is strictly valid and which
therefore can transmit stable waves, of either condensation or rarefaction, of
any amplitude, and (2) as we shall see in Section 4 it is the only fluid which
can support and transmit transverse
waves.
In general, the
adiabatic speed of sound waves c is
related to the pressure p and density
ρ by the equation
c^{2} = (dp/dρ)_{s
} (3.6)
From this, the
adiabatic speed of sound c ( i.e. no heat flow, ∆Q = 0)) in a
perfect gas is also
c^{2} = kpv = kp/ρ =kRT
(3.7)
By Eq. 3.6 a
positive wave speed c, requires that
dp/dρ must be positive. Since
v = 1/ ρ, we see that dp/dρ and dp/dv must have opposite
signs. Real gases, and the adiabatic
equation of state for the UF ( Tangent gas)
gas all have a negative slope for
dp/dv on the pv diagram and therefore
have positive adiabatic wave speeds.
For compressive
waves, the exact wave equation, expressed in terms of the wave function ψ
for amplitude, is [13]
Ñ^{2} ψ^{ } =^{ }1/c^{2}
∂^{2}ψ/∂t^{2 } ∕ [ 1 + Ñψ ]^{(k }^{+ 1)}
(3.8^{ })
or, for one
dimensional motion in the xdirection
∂^{2}ψ //∂x^{2} = [(1/c^{2}) ∂^{2}ψ /∂t^{2} ] ∕ [ 1 + ^{ }∂
ψ /∂x ]^{ 1+k}
(3.8a)
This exact
equation means that for real gases ( k
> 0) compressive waves are always unstable and grow with time . For very
small amplitude waves, however, the term in the denominator of Eqs. 3.8 and
3.8a involving ∂ ψ /∂x becomes approximately unity, and the equation
simplifies to approximately become the classical wave equation n for very low
amplitude sound waves (acoustic waves in real gases, which is
Ñ^{2 }ψ = (1/c^{2 })∂^{2}ψ/∂t^{2}
^{ }
and which has the general solution
ψ = ψ_{1(}x – ct) +
ψ_{2} ( x – ct)
In the case of the
UF, however we see that, since k = – 1, the exponent ( k + 1) in the denominator of Eqs. 3.8 and
3.8a becomes zero, thereby automatically reducing these equations to the simple classical wave
equation, but without any of the approximation needed for real gases such as
air.
The UF is
therefore truly unique in that it
automatically becomes exact for waves of any wave amplitude, large or
small and is no longer limited to
infinitesimal waves, as is the case with
real gases. The UF is therefore unique among gases, since in all
real gases finite amplitude waves always
either steepen to eventually form shocks or they die out, and only sound waves of infinitely low amplitude can persist as
stable waves [12,13].
The natural representation of the general solution of the
classical wave equation
ψ = ψ _{1}x – ct) +
ψ _{2}( x + ct)
is on the (x,t), ‘ physical plane’ or ‘spacetime’ diagram. Figure 3 shows the characteristic lines
representing two families of onedimensional leftrunning and rightrunning
waves in this’ spacetime’ or ‘physical plane’[12].
Figure3. Spacetime ( physical plane) Plot of Wave Characteristic Lines
(1dimensional )
For isothermal
motions ( i.e. constant
temperature, ∆T = 0), we have the
isothermal (Newtonian) speed of sound waves in a real perfect gas
c^{2} = pv =
p/ρ = RT ; c = [pv]^{1/2}
.
(3.9)
Now with our exotic gases we have
a. The
Tangent ( i.e. adiabatic) Equation of
State is: p = −Av + B
(3.10)
The general adiabatic sound speed equation is c^{2} = kpv, therefore, in the Tangent case we must
have
c^{2} =k pv = k [−Av^{2} +^{}Bv] = +Av^{2
}− Bv
(3.11)
and the sound wave speed c is positive as it should be.
b Our isothermal equation of state is: p = +Av − B
The wave speed
in an isothermal gas is given by
c^{2} = pv, and therefore we must have
c^{2} = pv = v[+Av − B]
= +Av^{2 }− Bv
(3.12)
Therefore, since the right hand side of
Eqn 2 is positive, the isothermal state also has a positive wave speed.
Moreover, since Eqs. 1 and 2 yield the
identical result (c^{2} = +Av^{2 }− Bv) then the
wave speed c is the same for both the adiabatic and isothermal states. This agreement of the two wave speed is
additional evidence that our adiabatic and isothermal equations for the k = −1 field are correct.
So far we have
not distinguished between longitudinal and transverse wave motions in the UF.
Clearly longitudinal waves are supported in the UF, and, moreover, they are
not restricted to low amplitude acoustic type waves as in real gases where k is positive. However, for the case where
adiabatic and isothermal motions are both
present we shall see that transverse fluid waves are also
uniquely supported in the UF (Section 4)
and that they in fact correspond to Maxwell’s electromagnetic waves.
The quantization of UF waves is also
presented in Sections 4 and 5.
3.5
Isentropic ratios ( c, p, ρ, T)
and relative motion (V)
The general
isentropic ratios relating pressure,
density and temperature for any gas [12] are ( using n = 2/(k−1))
c/c_{o} = [p/p_{o}]^{1/(n+2)} = [ρ_{o}/ρ]^{1/n}
= [T/T_{o}]^{1/2}
(3.13)
In the UF
where k = − 1 and t n =2/(k – 1) =
− 1, these isentropic ratios then become simply
c/c_{o} = p/p_{o} = ρ_{o}/ρ
= v/v_{o }= [T/T_{o}]^{1/2 }(3.14)
^{ }
Similarly, the
kinematic energy equation for relative motion (Eq. 3.5) with k = − 1 and n =
2/(k− 1) = − 1 becomes
^{ }
c/c_{o} = [ 1 + (V_{ }/c_{o})^{2}
]^{1/2}
(3.15)
Therefore, we
also have
p/p_{o} = [1+ V^{2}/c_{o}^{2}
]^{1/2}
p/p_{o }−1 = ∆p/p_{o} = [1+ V^{2}/c_{o}^{2}
]^{1/2} − 1
(3.16)
∆ρ/ρ = [1+ V^{2}/c_{o}^{2}
]^{1/2} − 1
(3.16a)
relating pressure
pulses ∆p, and corresponding density pulses ∆ρ to relative
fluid motions V in the UF, for example
to oscillations of an electric charge or accelerations of a material particle.
The various
possible types of wave motions can then be investigated by imposing (a)
condensation pulses ( +∆ρ), (b) rarefaction pulses (−∆ρ) and (c) density
oscillations ( ± ∆ρ) as in Fig. 4.
Figure 4. Density
Perturbations and Wave Motion in the Universal Field
Figure 5. Various Wave Motions in the UF
3.6
Static Values of Pressure, Temperature and Density for the UF
In any
compressible field the basic initial values of interest are often those there
is no fluid flow ( V = 0). These are called the static values and are
designated as p_{o, } T_{o} and ρ_{o}^{ }. Their
numerical values remain to be determined.
Having shown that
the Universal Field is a unique wave medium
supporting a variety of stable wave types and uniquely obeying the classical
wave equation without approximation , we now proceed to show how it can also
support transverse waves
corresponding exactly to Maxwell’s transverse
electromagnetic waves which transmit
light and radiation through space.
4,
The Universal Field (UF) Supports Transverse Waves Which Are fofrmally
Identical to Transverse Electromagnetic
Waves in Space
4.1 Real gases can only support longitudinal waves, that is waves
in which the density variations ±∆ρ
are along the direction of wave propagation. Real gases cannot support
transverse waves in which the density variations would be transverse to the
direction of wave propagation. Its was this inability to transmit the
transverse waves of light which led to the demise of the old luminiferous ether
concept. We now ask: What is the evidence for transverse fluid waves in the Universal Wave Field
(UF ) with its orthogonal adiabats and
isotherms?
Subjects
Evidence for transverse fluid waves
Maxwell/s electromagnetic waves
Polarization and spin
Quantization: Photons
We consider a simple pressure pulse ( ±∆p)
in the UF as in Fig.6 below:
Figure 6.
A pressure pulse ( ±∆p) in the Orthogonal Environment of the UF
The initial or
static state is designated as p_{o}._{ }When the pressure pulse
( +∆p) is imposed from outside in some way the UF must respond
thermodynamically in two completely orthogonal and hence two completely
isolated ways, namely, by (1) an adiabatic stable wave along the adiabat ( TG)
and (2) by an isothermal stable pulse along the isotherm (OG).
_{ }
Spatially, the
pressure disturbance ( +∆p) must
propagate in the direction of the initial impulse. But, since the two
components of the pulse are orthogonal, they must still remain completely
independent and physically isolated.
The only way possible for this to take place
is for the two mutually orthogonal components to also be transverse to
the direction of propagation of the two pressure pulses. Vectorially, this requires an axial wave
vector V in the direction of propagation
( say z) with the two pulses orthogonally
disposed in the xy plane. i.e. TG
X OG = V which is reminiscent of the
Poynting energy vector S = E x B in an electromagnetic wave.
E
A wave of amplitude ψ traveling in one
direction (say along the axis x) is represented
by the unidirectional wave equation
dψ/dx = 1/c dψ/dt
4.2 Maxwell’s electromagnetic waves
Here, however, in
the case of our adiabatic and isothermal pressure pulses we have two coupled yet isolated
unidirectional waves, and this reminds us of Maxwell’s coupled electromagnetic
waves for E and B, as follows
dE_{y}/dx = (1/c) dB/dt and dB_{y}/dx = (1/c)
dH/dt
where c is the
speed of light, E is the Electric intensity and B is the coupled magnetic
intensity.
Maxwell’s E and B vectors are also orthogonal to each
another and transverse to the direction of positive energy propagation.
Therefore, we have established in outline
a two component wave system in the
Universal Field (k = −1)
which formally corresponds to the E and
B two component orthogonal system of Maxwell for electromagnetic wave
propagation through space in a continuous medium. His equations for E and B
are
Curl E ∂ = −(1/c)
∂B/∂t
(4.1)
Curl B = (1/c) ∂E/∂t
(4.1a)
If we now
designate our Tangent gas as A ( for Adiabatic) and our Orthogonal gas as I (
for Isothermal) then our analogous wave equations would be
Curl A = − (1/c) ∂I/∂t
(4.2)
Curl I = (1/c) ∂A/∂t
(4.2a)
The two systems
are formally identical. Therefore, we propose that the medium in which
Maxwell’s transverse electromagnetic waves travel through space is physically identified with being a
Universal Compressible Field (UF) having the above described thermodynamic
properties for adiabatic and isothermal motions initiated in the UF by imposed
pressure pulses ( presumably by accelerated motions of electric charges.) The compressibility
of the UF now properly accounts on physical grounds for the finite wave speed ( speed of light), and
in addition wave motions in this
fluid medium are transverse as required by the electromagnetic
observations.
It is possible to
reduce Maxwell’s two equations UF equations to a symmetrical single wave
equation
∂^{2}E/∂x^{2 }=^{ }(1/c^{2}) ∂^{2}E/∂t^{2
} (4.3)
∂^{2}B/∂x^{2 }=^{ }(1/c^{2}) ∂^{2}B/∂t^{2}
(4.3a)
and similarly
with A and I for our Adiabatic/Isothermal coupled wave in
the UF:
∂^{2}A/∂x^{2 }=^{ }(1/c^{2}) ∂^{2}A/∂t^{2}
(4.4)
∂^{2}I/∂x^{2 }=^{ }(1/c^{2}) ∂^{2}I/∂t^{2}
4.4a)
This is not surprising
since the UF with its k = −1 thermodynamic property is the unique compressible fluid which automatically
generates the classical wave equation ( Eqn. 10) with its stable, plane waves.
The formal agreement of the UF theory with Maxwell is again striking.
Instead of taking our initial external perturbation as a pressure pulse ( +∆p) we should
more realistically from the physical standpoint take it to be a density
condensation (s = ( ρ – ρ_{o })_{ }/ ρ_{o}
= +∆ρ/ ρ_{o}).
This will now result in a positive pressure pulse (+∆p) appearing in the adiabatic (TG) phase of the UF but a negative pressure pulse ( −∆p) in the
isothermal or orthogonal perturbation component (OG) . This perturbation is
represented by the two orthogonal sets of arrows on the pv diagram, one
corresponding to +∆p and the other set corresponding to − ∆p.
As the wave progresses the two orthogonal vectors also rotate.
S
Figure 5. The physical ambiguity which
results from a pressure/density perturbation in the Orthogonal UF
An oscillating
density perturbation ( ±∆ρ) then results in an axial wave vector
having two mutually orthogonal
copmponents (A.I.) of a density perturbation wave. This appears to correspond formally to the Maxwell
electromagnetic wave system with its two mutually orthogonal vectors for
electric field intensity E and
magnetic field intensity B.
We have thus established a case for the
compressible UF being a cosmic entity
which transmits transverse electromagnetic waves through space. A
necessary next step will be to examine the UF in relation to all the other
established facts as to light and other radiations. These must include the
nature of electric charge, electrostatic fields, the compressed fields of moving
charges and the resulting magnetic fields, etc. etc. Preliminary work has
indicated that this additional reconciliation will be a complete and successful
one.
4.3 Polarization and Spin
Since we are dealing here with two linked
mutually orthogonal states, all the formal requirements of electromagnetic
polarization and spin are automatically satisfied. Various other
physical details remain to be examined by specialists.
4.4 Electromagnetic Wave Quantization: Photons and the Orthogonal
State Waves
Perhaps a bit simplistically we could just
consider each individual axial vector wave as single basic wave entity or
quantum entity and then build up more
complicated energetic states by
superposition of the basic linear
waves.
However, the quantization can be seen on a
more physical basis if we set the UF’s static or rest pressure p_{oi}
at some small value very close to p =
0., say at p_{o} = 6.673 x10^{11 }kilopascals. This at once
makes the maximum allowed value of the pressure fluctuation −∆p
equal to 6.67 x10^{11} kpa. as
well.
Now we ask what happens if the initiating
pressures ( density ) oscillation is of
greater amplitude then 6.67 x10^{11} kpa? Energy conservation is
achieved if the energy input driving the wave is divided into N quantized units
all of the same maximum permitted pressure amplitude ± ∆ ρ = ± ∆p
= N x const. = n x 6.673 x10^{11} kpa/., where N is a positive quantum
integer. This of course is just an application of the energy relation ε =
Nhν in quantum radiation theory. This quantization procedure is also
applied to gravitation in the succeeding section.
Figure
6. Forced quantization of large input density pulse +∆ρ by very
small static pressure level p_{o}
4.5 Special Relativity: The question of whether the Universal Field proposal reintroduces a cosmic “medium” into
space will also undoubtedly come up, although it is more a matter for
experiment to settle rather than for theory.
In the UF all motions are physically purely relative, which is to say
that the physical effects of motion
depend only on relative motion between the field and matter. In a compressible
field the laws to be tested are those of compressible motion and not those of classical incompressible motion which have
heretofore always been used in evaluating the MichelsonMorley and other
optical motion experiments ..
The kinematic energy equation (3.3) yields the Lorentz/Fitzgerald
transformation (Eq. 3.5) but in a more
general form than special relativity, and now on physical rather than on
postulation grounds. In the matter of the central problem of special relativity, which involves the direct
mathematical addition of material source velocities to the velocity of light,
the addition formula from compressible theory containing the energy partition
parameter n ( Eq. 3.5) must be used, and
not the failed approach of using the classical direct addition of velocities(
e.g. c + V and c −V) which
led special relativity being introduced
in the first place.
To repeat for clarity in this matter, the
new addition rule will involve (c ±V/√n)
instead of the old classical (c ± V).
From the kinematic energy flow
equation
c^{2} = c_{o}^{2} – (1/n) V^{2}
(3.3)
we get the ratio of wave
speeds
c/c_{o} = [ 1 – V^{2} / n c_{o}^{2}
]^{1/2}
(3.5)
which sets the
correct velocity addition formula for compressible flows. We see that it
involves the introduction of the thermodynamic energy partition parameter n
( n = 2/k – 1). In most cases
this addition of n reduces the expected
fringe shifts and oscillations changes predicted by the failed classical
formula and bring them into line with
experiment. For details see a current
review of the observational data of the MichelsonMorley and later experiments
at website www.energycompressibility.info
The formulae of
the Lorentz transformation, and hence of special relativity, are formally
identical to the above formulae for
compressible flow only in the special case of n = +1 ( k = 1.67) , for example in opnedimensional flow experiments on the apparent increase of mass with relative
velocity in accelerators. However, in
general, the value of n for any given
experimental system will not be unity,
since a relative motion
experiment involves not just the wave transmitting field but its
interaction with the motions and accelerations of matter.
Summary. Maxwell’s electromagnetic field equations
involve two mutually orthogonal field vectors E and B which obey the
classical wave equation and form a
transverse wave propagating in space. Maxwell’s system is a complete and
selfconsistent theoretical structure which then agrees with the facts of
experiment. The UF is the only known theoretical field ( k = n =
−1) which can support classical wave motion without approximation, and
which furthermore involves two mutually orthogonal field vectors A and I which can form and support a transverse wave propagating in the UF
through space. The two systems are formally identical, and, since Maxwell’s
equations are fully verified by experiment, then the UF system is also thereby
verified.
Finally, if
the UF can solidly establish itself as a
unique, compressible fluid medium which transmits transverse electric and
magnetic waves, then the question naturally arises: Can the compressible UF also account for the
unique nature of, and the spatial transmission of, the force of gravity? This is
discussed next..
5. The UF as a Support for the Transmission
of Graviational Force Through Space
Subjects
Exclusively attractive nature of gravitational
force
Extreme weakness of gravitational force
A unique wave speed associated with
gravitational and other UF waves
Gravitational Characteristics: The principal characteristics of
gravitational force to be properly
accounted for in a new theory are (1) its exclusively attractive nature for
mass, (2) its extreme weakness relative to the electromagnetic force, and (3)
its 1/r^{2} decline in strength with distance.
Any new,
physically based theory attempting to explain the nature and behaviour of
gravitation will obviously have to account for
the three physically characteristics set out above. It will not have to
meet the general relativity postulates, but it will eventually have to explain
why the latter theory does make successful
predictions which offer corrections to the Newtonian predictions, such
as the advance in the perihelion of Mercury, and various gravitational lensing
effects on light waves.
5.1 Exclusively
attractive forces in the Universal Field
In any
compressible fluid force is given by the Euler equation , which for 1dimensional flow is
F = ∂u/∂t + u ∂u/∂x
+ v ∂u/∂y +w ∂u/∂z = −(1/ρ) ∂p/∂x
(5.1)
where the term on the right hand side is called the pressure
gradient force.
In any
compressible fluid medium, waves set up local transient pressure forces. Most waves in real gases and
fluids are pressure oscillations (±∆p) and so they do not exert any net directional force on a
material object they encounter. However,
in the UF special types of waves can occur which can be either exclusively
pressure compressions (+∆p) thereby exerting a net repulsive force on any
material body in their path, or they can
be exclusively pressure rarefactions (−∆p) which would then exert exclusively attractive force.
Consider Figure 6 where a condensation pressure pulse (+∆ρ; +∆ρ) imposed on the initial static pressure p_{o
} in the UF will produce a
rarefaction pressure pulse (−∆ρ) in
the isothermal mode of response. Consequently, a source of pressure
condensations will produce a train of isothermal pressure pulses as a response
in the UF, and these pulse will travel spherically outwards through space.
When these rarefaction waves eventually impact a material body
(mass) they will exert a net attractive
force on it. This mechanism therefore in simple outline is formally equivalent
to the force of gravity being produced by a rarefaction pressure gradient
force.
Figure 7. Positive density pulse ( +∆ρ) of any magnitude produces a quantized
gravitational rarefaction pulse (−∆p_{g} ) of constant magnitude p_{g} =6.67 x
10^{11} kilopascals
5.2 Extreme
Weakness of the Gravitational Force and the Universal Field
The three main
forces in nature are the strong nuclear force, the electromagnetic force and
the gravitational force. The nuclear force is much the strongest. The electromagnetic force is
about 1/137 th of the nuclear force. The
force of gravity, however, is incredibly weaker that the other two, being only
about 10^{40} th of the strength of the electromagnetic
force. Thus we have
F_{s} / F_{e/m} = 1/137 ;
and F_{g} / F_{e/m} = 10^{40}
We have shown
above that a wave train exclusively made up of pressure rarefactions would
explain the attractive nature of gravity. Now we must explain how these
rarefaction waves can all be so extremely weak.
Consider a
positive density pulse (+ ∆ρ) imposed on the UF at some point O
(Fig.6). Adiabatically this will result
in a pressure increment (+∆p) ,
but in the isothermal mode this will give a pressure rarefaction (−∆p).
We must now consider whether a sufficiently
large negative pressure pulse (−∆p) can transmit in the
isothermal mode right through the p = 0
point of expansion and on into the negative pressure region of Quadrant IV. It
appears that it cannot, because of the fact that a negative pressure will
entail a negative temperature, and quantum theory and experiments show that temperature of opposite sign are not
equal in magnitude [a, b, c] Thus the isothermal condition will be
discontinuous on the v=axis at the p = 0 point. Physically this appears to require that the
negative pressure pulse must terminate at this point. The consequence then is
that all negative isothermal pressure
pulses will be of the same amplitude,
(−∆p) _{isothermal} =
( p_{o} – 0) = p_{g} – 0
= constant = 6.67 x 10^{11 } kilopascals.
Therefore, if the
initial pressure/density perturbation is
inserted into the UF at its static pressure p_{g} = 6.67 x 10^{11}
kpa, then the induced negative
isothermal pressure rarefaction (−∆p ) will be (a) extremely
weak and always of the same amplitude
of 6.67 x ^{10}11 kpa. no matter what the amplitude of the
initiating positive density pulse (+ ∆ρ).
Again, we have F_{g} = − v dp/dx = − 1/ ρ
dp/dx . Therefore, for unit mass at unit
distance we have F_{g} = − ∆p = G = 6.673 x 10^{11}. Thus the basic UF static pressure p_{o}
seems to correspond numerically to G, the gravitational constant, as we
have just postulated (p_{o} =
6.673 x10^{11} kilopascals).
On this physical
model, gravitational force is carried through space isothermally by a train of
rarefaction pressure pulses (−∆p), all physically constrained to
have the same invariant amplitude regardless of the amplitude of the
initiating / density pulse (+ ∆ρ).
Now the
gravitational force is known to be around
10^{40} th of the strength of the electromagnetic force when
the latter is set at unity. Then to
show that our gravitational formulation gives the required weakness for the gravitational
force, consider
F_{ig} =( G m_{1} m_{2}) / r^{2
} (5.2)
and the Euler
pressure gradient force
F_{p.g} = −1/ρ (dp/dx) = −v
(dp/dx)
(5.3)
If we now take dp
= lGl and, realizing that the equation is for unit mass ( m = 1) , we get (at
unit distance dx = 1) the force
F_{g } = − (1) 6.67 x 10^{11 }
F_{g }^{ }= 10^{28} [ 6.67 x 10^{11}]
= 6.67 x 10^{39}
While this is only a rough calculation, it shows that the
theory yields the required general magnitude for the strength of the
gravitational force.
5.3 A Unique Wave Speed Assciated with Gravitational Waves
In general, in the UF the wave speed, given by c^{2}
=(3 x 10^{8})^{2 } = ∆p
/∆ρ holds, and the wave speed c is the speed of light in space .
However, consider what happens when in the generation of a gravitational wave
pulse as just described above, the
reduction in p in the isothermal mode approaches the zero pressure point. At this point, if the wave pressure action
then cannot continue on into negative values ofs p in Quadrant IV as required
by the magnitude of the initiating positive pressure pulse , then we have the UF fluid approaching a state where v = 1/
ρ = constant and so any
additional ∆ρ needed must become zero. But then the gravitational wave speed must approach
infinity since c^{2} = dp/ 0 must then equal infinity.
As to the probable magnitude
of this new gravitational pulse wave speed we propose the following: The ratio of the
gravitational constant to that of the Planck constant is
G/h = 1.04 x 10 ^{23} =
constant
(5.4)
If mass is taken as
dimensionless ( as we do everywhere in this theory on the grounds that mass is
a condensation of energies and so can be
consider a ratio of energy before and after elementary particle formation ) then the dimensions of the ratio G/h are those of velocity or speed [l t^{1}].
It thus seems possible that the ‘reflection’ of the
positive pressure waves of Quadrant I at
the p = 0 boundary or discontinuity may generate an additional transient,
extremely fast UF wave as well as propagating
the basic electromagnetic or gravitational pulses that it reflects and
sustains.. This new wave seems more likely to relate to the spread of quantum
information through space rather than to the transfer of energy.
In summary, we have exclusively attractive gravitational
waves traveling as the speed of light speed of light, but at the same time as
each wavelet reaches the zero pressure point ut emits a secondary wave which
spreads spherically at quasiinstantaneous speed.
For a variety of reasons it is reasonable to associate this
secondary quasiinstantaneous pulse radiation with quantum wave information
spread.
Summary. We have shown that the UF can support and
transmit stable rarefaction waves which
(1) exert exclusively attractive force on masses and (2) have the necessary
extreme weakness. This meets the general requirements for them being
gravitational waves and for the UF
therefore being the physical seat of universal gravitational force.
There are, of course, many other aspects of gravitation we
have not considered here These also will have to be considered in light of
the theory, but they are not essential
for this general presentation.
6. The UF as a Support for the Transmission
of Quantum Information Through Space
Subjects
Compressibility and Quantum Mechanics
Quantum Wave Function
Normalized Quantum Wave Function
The Extra Energy term 2cV and Fundamental Quantum
Relationships
Mass Ratios of the Elementary Particles of Matter
Quantum Wave Speed, Entanglement and Collapse of the Wave
Function Problem
While this
subject has not yet been studied in great detail, there are some aspects which already indicate that
compressible flow and the UF concept are intimately related to quantum phenomena, just as has been shown
above for electromagnetism and gravitation.
The standard model of quantum physics is one of the
most remarkable achievements of science.
It explains an enormous range of nuclear and atomic phenomena to a very
high degree of accuracy, and is logically quite consistent. Yet, for all its success, it has some serious
deficiencies.
For example, while it is very successful in describing
many aspects of particle creations, annihilations and interactions, it still
has no predictive power to specify the values of the observed masses, and the
massratios of the elementary particles – the experimentally determined values
of these masses must still be put into the theory by hand. Again, it requires the arbitrary introduction
of various fundamental constants, such as the speed of light c, and the fine
structure constant of the atom α, which are essential for its
calculations, but for whose physical existence or numerical value it has no
explanation whatsoever. Again, quantum
physics is still essentially unrelated to classical mechanics, and although it
can be related to electromagnetic theory through quantum electrodynamics, this
is so only for the cases of the electron – the phenomena of the nucleus of the
atom are as yet still essentially unrelated to the rest of physical science in
the standard model of quantum theory.
Perhaps most seriously it has no explanation for the physical nature of
the basic quantum wave function, nor for the transfer through space of quantum
information, nor for the famous problem of the ‘collapse of the wave function’,
nor for the wave/particle duality of matter. Yet again, quantum physics has
little relationship to relativity, gravitation and cosmology.
Here we shall touch on only a few key aspects as an indication of the relevance of the
compressible field (UF) theory to quantum physics.
6.1 Compressibility and Quantum Mechanics
We shall take the position that all quantum phenomena
are basically compressible energy flow manifestations. The basic energy form ( variously to be called a
characteristic, ray, wave pulse, wavelet, etc.) is the quantum wave function
ψ which refers to a single, compressible, enray pulse or wavelet.
More complex waves ( elementary particles, etc. ) are
built up from the linear enray ψ by superposition in a irreversible shockwave compression event.
1. The unsteady
flow ( wave pulse) energy equation is, as shown above
c^{2} = c_{o}^{2} – (1/n) V^{2}
– 2cV/n
(6.1)
where the extra
energy term 2cV/n is the result of the pulse or acceleration of relative
motion. For the electromagnetic case tthis could refer to the energy of the
acceleration or oscillation of a charge which generates the electromagnetic
waves. All the other basic quantum
relations such as the de Broglie wave
/particle equation p = h/λ,
the Heisenberg uncertainty relationship,
the quantum operators for position and momentum and so on, can also be derived from this 2cV or ‘extra energy “ term. (
see below)
6.2 Basic Quantum Wave
The wave function ψ is the linear, ‘characteristic’, or
‘ray’ solution, of the hyperbolic, linearised, approximate differential
equation called the classical wave equation
Ñ^{2}ψ =
1/c^{2} ∂^{2}ψ/∂t^{2}
; ∂^{2}ψ/∂x^{2}
= 1/c^{2} ∂^{2}ψ/∂t^{2 }(6.2)
Therefore, the basic formula for the characteristic or
ray is as follows:
Ψ = c ± V
This may
also be complex, as Ψ
= c ± iV
6.3 Quantum Wave Function
ψ = c ±
V (6.3)
Here, V is the relative velocity and may be set to
zero, making c = c_{o} in the ‘at rest’ coordinate system chosen. (cf.
Sects. 2.8; 5.2). For an energy flow, c_{o} is 3 x 10^{8}m/s.
In general, ψ is complex, and we then have
ψ = c ±
iV
(6.4)
and
ψ^{2} = c^{2} – 2icV +V^{2}
The physical nature of the quantum wave function is
thus the relative flow velocity V ( or
particle momentum, mV = p)
plus the wave velocity c (or wave momentum, mc).
This still leaves open the question of the physical nature of the wave and the wave
field, which however have been dealt with above where the classical waves of
the Universal Field (UF) ( n and k both equal to –1) were considered both for electromagnetism and for
gravitation.
6.4 Normalized Quantum Wave Function
From the kinematic energy flow equation (c/c_{o})^{2} = 1 – 1/n (V/c_{o})^{2}, we have the normalized wave function
Ψ_{N }= c/c_{o} + i V/c_{o}; ψ*_{N }= c/c_{o} –
i V/c_{o
}(6.5)
For small V, this reduces to ψ_{N} ≈
c/c_{o}.
This wave speed
ratio c/c_{o} relatesof course,
to all the isentropic,
thermodynamic ratios p/p_{o},
ρ/ρ_{o}, T/T_{o } as
c/c_{o} = [p/po]^{1/(n+2)} = [ρ/ρ_{o}]^{1/n} = [T/T_{o} ]^{1/2}
For the UF where k
= – 1 = n, we have
c/c_{o}
= p/p_{o} = ρ_{o}/ρ = [T/T_{o}]^{1/2}
6.6)
The Fitzgerald/Lorentz contraction factor in the
general case is now c/c_{o} = [ 1 – (1/n) (V/c_{o})^{2}
]^{1/2 }. For the UF where n itself is negative, this has the
minus sign reversed to become
c/c_{o}
= [ 1 + V^{2} /c_{o}^{2} ]^{1/2}
(6.7)
a new Lorentz relationship which requires special
study.
6.5 The ‘Extra Energy’ Term, 2cV yields the Following Fundamental Quantum
Relationships:
The unsteady energy equation
c^{2} = c_{o}^{2} – (1/n) V^{2}
– 2cV/n
(6.8)
contains a wave pulse energy term 2cv.
Here we show that this then yields the fundamental quantum relationships as
follows
a) Planck’s Constant h
For n = 1, if cV = constant energy for each waves, then cV/υ = constant energy per
cycle or pulse:
cV/υ
= h
cV =
hυ = hω/2π = ħω = ε_{υ
}(6.9)
For the complex case,
cV/υ
= ħ/I = iħ
b) De Broglie Wave /Particle Equation
cV/υ
= h
But c/υ =
λ; V(m) = p (momentum), so
λp = h, or
p =
h/λ
(6.10)
c) Lagrangian Function, L
L =
2cV
d) Quantum Wave Function Operators :
1) Hamiltonian Energy Operator
cV=
hυ = ħω = ε
(6.11)
icV = iħω
But
iω = ∂../∂t, and so cV = h/I ∂../∂t = +iħ∂../∂t = H_{op}
which is the Hamiltonian energy operator.
( To ensure correct dimensions, it must be applied to
the normalized quantum function ψ_{N}).
2) Momentum
cv = hυ =
+hω = ε
V = (1/c))ħω,
or (m)V = p = (m)(1/c)ħω
Multiplying by i, we have:
(m)iV = (m)(1/c) iħω
= (m)(1/c) ħ
∂../∂t
So, we have
(m) V = p = (m)(1/c) iħ ∂../∂t
But,
I1/c) ∂../∂t
= ∂../∂x, and so
(m)V = p = iħ∂../∂x = p_{op
}(6.12)
which is the quantum wave operator, ( to ensure
correct dimensions, it must be applied to the normalized quantum function
ψ_{N}).
e) Heisenberg Uncertainty Principle
cV =
hυ; cv/υ = h
λV = h
But λ = Δx and V(m) = Δp, so
Δx .
Δp ≥ (m) h
(6.13)
which is the Heisenberg uncertainty principle.
6.6 MASS RATIOS OF ELEMENTARY PARTICLES
It has been proposed that all elementary particles of
matter (with the possible exception
of the neutrino) are condensed
energy forms. The forms are given in terms of a simple, integral number n (
n = degrees of freedom of the compressible energy flow):
Baryons and
Heavy Mesons
For the baryons and heavy mesons the energy condensation that produces the mass is postulated to
take place via the strong shock option [
] and is proportional to the shock strength given by = [n+1]^{1/2}
m_{b}/m_{q}
= V_{max}/c* = [n+1]^{1/2
}(6.14)
m_{b} is the mass of any baryon particle, m_{q}
is a quark mass, V_{max} = c_{o} n^{1/2} is the escape
speed to a vacuum; that is, it is the maximum possible relative flow velocity
in an energy flow for a given value of n, the number of degrees of freedom of
the energy form, This is a
nonisentropic relationship, and it corresponds physically to the maximum
possible strong shock.
Experimental evidence for this new baryon mass ratio
formula is given in the following Table :
Experimental
Verification of the Mass Ratio for Baryons and Heavy Mesons

n n +1 [n+1]^{1/2 }Particle Mass (m_{b}) Ratio to
(MeV) quark mass
0 1 1 quark (ud) 310 MeV 1
(s) 505
1
2 3 1.73 eta (η) 548.8 1.73^{ }
3
4
5 6 2.45 rho (ρ) 776 2.45
6
7
8 9 3 proton (p) 938.28 3.03
(1)
neutron (n) 939.57 3.03
Λ (uds) 1115.6 2.97
(2)
Ξ^{o}
(uss) 1314.19 2.99
(3)
9 10 3.16 Σ^{+} (uus)
1189.36 3.17
(2)
10 11 3.32 Ω^{} (sss) 1672.2 3.31
(4)
Note: Average quark mass is 310 MeV; (2) Average quark
mass is (u + d+ s)/3 = 375 MeV (B)
Average quark mass is (u+s+s)/3 = 440 MeV; (4) Average quark mass is 505 Mev.
Therefore, Equation 21 is verified to within about 1%.
Note:
For the UF with k = − 1 = n, shocks are impossible, since V can approach
but never equal c [ ] and the Mach
number M = V/c never reaches or exceeds unity, as required for condensation
shock formation . Also, we see that in the UF
[ n + 1]^{1/2 } becomes
zero, again confirming that no mass condensation of flow energy can take place
in the UF. Thus, the origin of the
energy condensation which is our
postulated origin for the emergence of mass takes place entirely in
World A where n is positive.
Leptons,
Pion and Kaon
We form the
ration of the mass of each lepton m_{L}
to the mass of the electron m_{e}
as
m_{L}/m_{e}^{
} =
k/α^{2} = [(n+2)/n]/α^{2} = {(n+2)/n] x
137 (6.15)
where α = 1/11.703 is the fine structure
constant, and k is the adiabatic exponent or ratio of specific heats, k = c_{p}/c_{v}
= [(n+2)/n]. Because of the presence of k, this formula for the mass of the leptons
is a thermodynamic and quasiisentropic one.
The leptons are formed via the weak shock option.
The experimental evidence for the lepton mass ratio
formula is given in Table below.
Lepton Mass ratios
n k =
(n+2)/n Particle Mass Ratio Ratio
(MeV) to x 1/137
Electron
1/3
7 Kaon
K^{±}
493.67 966.32 7.05
2
2 Pion π^{±} 139.57 273.15 1.99
4
1.5 Muon μ 105.66 206.77 1.51

 Electron 0.511 1
Clearly, k ≈ m_{l}/m_{e}
(1/137), supporting Equation (22).
6.7 Quantum Wave Speed,
Entanglement and Quasi instantaneous Collapse of the Wave Function Problem
A examination of these quantum subjects indicates that a
quasiinstantaneous transfer of quantum information would remove many of the difficulties and some
socalled quantum weirdness, including action at a distance problems. It is natural then to propose some
connection between the
quasiinstantaneous secondary
wave pulses we have discussed at the p = 0 discontinuityand which arise
witrh all UF waves if we set p_{o}
the static pressure at 6.673 x 10^{11}
, that is to say so near to the zero pressure point that all quantum wave
oscillations automatically must reach the zero pressure point and be reflected
and quantized as we have described above.
The possibility of a secondary wave of quasi instantaneous speed
being generated then emerges. This secondary, quasi instantaneous wave is
proposed as a quantum information wave. It appears suitable for the
transfer of information at superluminal
but not quite instantaneous speeds, and so able to deal with the quantum theory
problems such as those associated with
action at a distance, quantum entanglement
and collapse of the wave function. and others..
SUMMARY: We
have been able to related the fundamental quantum relationships to a single
energy pulse term 2cV/n. in compressible flow theory.
We have, in effect, quantized the various energy
‘fields’ represented by 2cV/n for various values of n, by equating them to the
‘timelike’ condition set by the frequency υ in the quantum equation
hυ = 2cV/n.
(Note that these equations, as is usual in
compressible flow theory, are for ‘specific’ energy, that is, for unit mass flow. For a definite particle, the
numerical value of the mass is to be inserted  the dimensions of the equations being not thereby changed, since in our system,
mass (m) is dimensionless. Thus, for the photon, we have hυ = m_{γ}
cV, where m_{γ} is the relativistic mass of the photon. In terms
of the photon momentum, we have
hυ = (m)cV
= cp
(6.16)
which is the de Broglie equation for the photon.
Other fundamental difficulties with quantum theory may also be removed by the introduction
of the new quasiinstantaneous speed for
the transfer of quantum information through space..
7. Experimental Evidence
for the Existence of the Universal Field
The above Sections have presented mostly theoretical evidence for
the reality of the UF.
Naturally, verification of the theory will also require
experimental evidence. . The mass ratio
evidence in Section 6 is one such set of
experimental evidence. Another is the well established optical shifts related to relative motion and
accelerations of the MichelsonMorley, Fizeau and Sagnac type experiments Here,
Lorentz himself was of the opinion that any optical effect whatever, such as a
fringe shift of any magnitude constituted
disproof of special relativity and he quoted Einstein to back up his opinion.
A new experimental approach emerges as a consequence of the new
orthogonal isothermal equation of state. This isothermal state requires a flow of heat (∆Q) to accompany any UF wave activity.
Some of this heat may possibly flow to the UF from our
real physical world A, in which case temperature fluctuations should in
principle be detectible. Preliminary, but quite extensive, experiments carried
out nearly two decades ago did detect temperature fluctuations apparently
linked to the inertial forces accompanying mass acceleration. These previously
inexplicable findings are currently being reexamined in the light of the new
UF wave theory.
References
1, E.T Whittaker, History of the Theories of the Aether and Electricity, 2 vol., 2nd ed.
2. Bernard A. Power, Shock Waves in a Photon Gas. Contr. Paper No. 203, American Association
for the Advancement of Science, Ann. Meeting,
3.,
NASA Proposal: Control No. K 2453;
Date:,
4. , Unification of
Forces and Particle Production at an Oblique Radiation Shock Front. Contr. Paper N0. 462. American
Association for the Advancement of
Science, Ann. Meeting, Washington,
D.C., Jan 1982.
5.
, Baryon Massratios and Degrees of Freedom in a
Compressible Radiation Flow. Contr. Paper No. 505. American Association
for the Advancement of Science, Annual Meeting, Detroit, May 1983.
6.
A. Kamenshchick, U. Moschella, and
V.Pasquier, Phys. Lett. B 511 (2001) 265268.
7. N. Bilic, G.B. Tupper and
R.D.Viollier. Unification of Dark Matter
and Dark Energy: the Inhomogeneous Chaplygin Gas. Astrophysics,
astroph/0111325, 2002.
8. P.P. Avelino, L.M.G. Beca, J.P.M de
Carvalho, C.J.A.P. Martins and P.Pinto. Alternatives to quintessence model
building. Phys. Rev. D.67 023511, 2003.
9. N. A. Bachall, J.P. Ostriker, S. Perlmutter, P. J.
Steinhatrdt, Science, 284 ( 1999) 1481.
10. S. Chaplygin, Sci. Mem., Moscow Univ. Math. Phys. 21 (1904) 1.
11. H.S. Tsien,
J. Aeron. Sci. 6 (1939) 399.
12.
T. von Karman, J. Aeron. Sci. 8 (1941)
337.
13. Horace Lamb, Hydrodynamics. 6^{th} ed (1936) Dover Reprint,Dover Publications
Inc. New York.
14.
A. H. Shapiro, The Dynamics and
Thermodynamics of Compressible Fluid Flow. 2 Vols. J. Wiley & Sons, New
York, 1953.
15.
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves.
Interscience , New York, 1948.
Copyright 2005 Bernard A. Power [Consulting
meteorologist (ret.)]